Conformal Fractional Dirac Operator and Spinorial Q-curvature

TL;DR

This paper introduces a new conformal fractional Dirac operator, explores its associated Yamabe problem, and establishes related inequalities and curvature notions, advancing the understanding of fractional spinorial geometric analysis.

Contribution

It defines the conformal fractional Dirac operator, develops a Caffarelli-Silvestre extension, and introduces a spinorial Q-curvature, extending classical geometric concepts to fractional settings.

Findings

Established a Caffarelli-Silvestre type extension for the fractional Dirac operator.

Derived energy inequalities and Sobolev inequalities for spinors.

Introduced a spinorial Q-curvature generalizing classical scalar Q-curvature.

Abstract

In this paper we introduce the conformal fractional Dirac operator and its associated fractional spinorial Yamabe problem. We also present a Caffarelli-Silvestre type extension for this fractional operator, allowing us to express it as a Dirichlet-to-Neumann type operator. As a consequence, we exhibit energy inequalities associated to this operator along with a weighted type Sobolev inequality for spinors. In the second part of the paper, we focus on the critical operator (which can be local or non-local depending on the evenness of the dimension). We introduce a Q-curvature operator, acting on spinors generalizing the classical notion of the scalar Q-curvature.